To be honest, looking at advanced stats as a whole (i.e. all of the big ones) is similar to the eye test you mention. Take PK for example... he was good during the first half of the season, but he was definitely playing below his standards. And it reflected in his stats. Since 2015, and especially in the month of February, he's been playing out of his mind and anyone who watches the games can see it. Lo and behold, he's back at the top of the league in terms of advanced stats (which really buckle down to simply shot differentials). Similarly, the couple of times I've seen Giordano play, he blew me away. Whenever he'd be on the ice, the Flames would be in possession and would carry the play. Same thing goes for Letang and Doughty, both of whom have incredible possession stats this year.
If I were able to watch every single game of every single team, then I wouldn't need to look at possession stats, as I'd have a good idea of what the top D in the league can offer in terms of possession. Unfortunately, we all have lives outside of hockey (or we'd be analysts and not forum dwellers). Luckily, we can use possession stats to get an idea of how one player compares to another. And when one player completely dominates an other player in just about every stat in the book, you can make the assumption that the former is playing better hockey than the latter... in this case PK vs Weber.
I'll agree that using possession stats exclusively to rate certain players is dumb, especially when you start looking at individual stats out of context. You need to find a balance, or you end up like Berkshire who seemed to think Diaz was a stud.